Respuesta :
Answer:
Approximately [tex]1.4 \times 10^{17}\; {\rm m}[/tex].
Explanation:
Assume that the distance between this star and the Earth is [tex]r[/tex]. For an observer on the Earth, it would appear as if the total power output (luminosity, [tex]L[/tex]) of the star was evenly distributed over the area of an imaginary spherical shell centered at the star, and the Earth would be a point on that spherical shell. The radius of this imaginary sphere would be the same as the distance [tex]r\![/tex] between the star and the Earth.
The apparent brightness of the star for that observer on the Earth would be equal to the power received across each unit area of this imaginary sphere:
[tex]\begin{aligned}& (\text{apparent brightness}) \\ =\; & \frac{(\text{luminosity})}{(\text{area of imaginary sphere})} \\ =\; & \frac{(\text{luminosity})}{4\, \pi\, r^{2}}\end{aligned}[/tex].
Rearrange this equation and solve for the distance [tex]r[/tex]:
[tex]\begin{aligned}r &= \frac{1}{2}\, \sqrt{\frac{(\text{total out}\text{put power})}{(\text{apparent brightness})\, \pi}} \\ &= \frac{1}{2}\, \sqrt{\frac{4.8 \times 10^{25}\; {\rm W}}{2.0 \times 10^{-10}\; {\rm W\cdot m^{-2}}}} \\ &\approx 1.4 \times 10^{17}\; {\rm m}\end{aligned}[/tex].
Final answer:
Using the luminosity distance formula, we calculate that the star with a luminosity of 4.8 x 10²⁵ watts and an apparent brightness of 2.0 x 10⁻¹⁰watt/m² is approximately 14.69 light-years away from Earth.
Explanation:
To find the distance to a star using the luminosity distance formula, you can use the relationship F = L / (4πd₂), where F is the apparent brightness, L is the luminosity, and d is the distance to the star. Given that the luminosity (L) of the star is 4.8 × 10²⁵ watts and the apparent brightness (F) at Earth is 2.0 × 10⁻¹⁰ watt/m², we can rearrange the formula to solve for d: d₂ = L / (4πF). Substituting the given values yields:
d2 = (4.8 × 10²⁵) / (4π × 2.0 × 10⁻¹⁰)
First, calculate the denominator: 4π × 2.0 × 10⁻¹⁰ = 2.5 × 10⁻⁹
Next, divide the luminosity by this value to get d₂:
d₂ = (4.8 × 10²⁵) / (2.5 × 10⁻⁹) = 1.92 × 10³⁴
Then take the square root of both sides to find d:
d = √(1.92 × 10³⁴) = 1.39 × 10¹⁷ meters
To convert this distance into light-years (ly), you would divide by the number of meters in a light-year:
d = 1.39 × 10¹⁷ meters / 9.46 × 10¹⁵meters/ly ≈ 14.69 light-years
Therefore, the distance to the star is approximately 14.69 light-years.
